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One of applications of “slope” to explain puzzles and paradoxes -
Triangle Dissection Paradox

"Below the two parts moved around - The partilisions are exactly the same, as those used above - From where "come" this hole?"

Explain: In the figure, the slope of the “hypotenuse” in figure 1 and figure 2 are completely different. (Click on image to see full size).

Also, The above two figures are rearrangements of each other, with the corresponding triangles and polyominoes having the same areas. Nevertheless, the bottom figure has an area one unit larger than the top figure (as indicated by the grid square containing the dot).The source of this apparent paradox is that the “hypotenuse” of the overall “triangle” is not a straight line, but consists of two broken segments. As a result, the “hypotenuse” of the top figure is slightly bent in, whereas the “hypotenuse” of the bottom figure is slightly bent out. The difference in the areas of these figures is then exactly the “extra” one unit. Explicitly, the area of triangular “hole” (0, 0), (8, 3), (13, 5) in the top figure is 1/2, as is the area of triangular “excess” (0, 0), (5, 2), (13, 5) in the bottom figure, for a total of one unit difference. Source: Triangle Dissection Paradox on Mathworld.wolfram.

  • Slope:  In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m. Slope is calculated by finding the ratio of the “vertical change” to the “horizontal change” between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient (“rise over run”), giving the same number for every two distinct points on the same line. The rise of a road between two points is the difference between the altitude of the road at those two points, say y1 and y2, or in other words, the rise is (y2 − y1) = Δy. For relatively short distances - where the earth’s curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is (x2 − x1) = Δx. Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line.
  • Also, here are the direction of a line is either increasing, decreasing, horizontal or vertical:

        - A line is increasing if it goes up from left to right. The slope is positive, i.e. m>0.
       -  A line is decreasing if it goes down from left to right. The slope is negative, i.e. m<0.

       -  If a line is horizontal the slope is zero. This is a constant  function.
       -  If a line is vertical the slope is undefined (see below).

    The steepness, incline, or grade of a line is measured by the absolute value of the slope. A slope with a greater absolute value indicates a steeper line - Source: Slope on Wikipedia


via Reddit user cgibbard: “The first transformation is the classic hinged dissection of an equilateral triangle into a square popularised by Dudeney.

The Wallace-Bolyai-Gerwien theorem shows that any two polygons with equal area must admit a dissection into finitely many pieces where one is allowed to arbitrarily rotate and translate the pieces to go from one polygon to the other. The problem about whether a hinged dissection exists remained open until 2007. You can read the paper here, which presents a method which always works to find a hinged dissection.”


“It seems that the more I tried to make my life about the pursuit of art, the more money controlled my life: collecting unemployment insurance, the humiliation of borrowing money from friends and family, tossing and turning at night while trying to figure out how to pay the rent. To survive I had to work hard jobs and afterwards I’d feel too tired and too stressed to paint. It’s very hard to create under those circumstances. Creativity is a delicate process. Often times I wonder if I should have just pursued a career for the first half of my life, obtained some degree of financial security, and then transitioned into art.”

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